Single word for the process of giving/getting a blood sample

'Extraction' might be a good fit for a direct opposite of injection, though it's most commonly used to refer to tooth removal. From https://www.merriam-webster.com/dictionary/extraction:

1:  the act or process of extracting something

And,

3:  something extracted

On the other hand, perhaps 'draw' is the word you seek. I would use it before I'd use extraction; though a cursory dig yielded no formal support for the use of 'draw' as a noun in this specific way.

The nurse gave an injection.

The nurse took a draw.

Phlebotomy is the act of drawing or removing blood from the circulatory system through a cut (incision) or puncture in order to obtain a sample for analysis and diagnosis.

So, phlebotomize - To perform phlebotomy upon (OED).

1596 T. Nashe Haue with you to Saffron-Walden sig. Bv, Phlebothomize them, sting them, tutch them. (OED)

2002 Amer. Jrnl. Gastroenterol. 97 1095/1 This reviewer phlebotomizes all nonacutely ill HCV [= hepatitis C virus] patients with serum ferritins of >200 ng/ml by removing a unit of blood every other week.

I would suggest venipuncture:

In medicine, venipuncture or venepuncture is the process of obtaining intravenous access for the purpose of intravenous therapy or for blood sampling of venous blood.